Space-time fractional stochastic partial differential equations driven by Lévy white noise

This paper is concerned with the following space-time fractional stochastic nonlinear partial differential equation {equation*} (∂_t^β+ν{2}(-Δ)α/ 2) u=Iₜ^γ[ f(t,x,u)-∑ᵢ₌₁ᵈ ∂/∂ x_i q_i(t,x,u)+ σ(t,x,u) Ft,x] {equation*} for a random field u(t,x):[0,∞)ᵈ, where α>0, β∈(0,2), γ≥0, ν>0, Ft,x is a Lévy space-time white noise, Iₜ^γ stands for the Riemann-Liouville integral in time, and f,qᵢ,σ:[0,∞)ᵈ are measurable functions. Under suitable polynomial growth conditions, we establish the existence and uniqueness of L²(Rᵈ)-valued local solutions when the Lévy white noise Ft,x contains Gaussian noise component. Furthermore, for p∈[1,2], we derive the existence and uniqueness of Lᵖ(Rᵈ)-valued local solutions for the equation driven by pure jump Lévy white noise. Finally, we obtain certain stronger conditions for the existence and uniqueness of global solutions.